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A bridge in a graph is an edge whose removal disconnects the graph and increases the number of connected components. We calculate the fraction of bridges in a wide range of real-world networks and their randomized counterparts. We find that real networks typically have more bridges than their completely randomized counterparts, but very similar fraction of bridges as their degree-preserving randomizations. We define a new edge centrality measure, called bridgeness, to quantify the importance of a bridge in damaging a network. We find that certain real networks have very large average and variance of bridgeness compared to their degree-preserving randomizations and other real networks. Finally, we offer an analytical framework to calculate the bridge fraction , the average and variance of bridgeness for uncorrelated random networks with arbitrary degree distributions.
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In network science complex systems are represented as a mathematical graphs consisting of a set of nodes representing the components and a set of edges representing their interactions. The framework of networks has led to significant advances in the understanding of the structure, formation and function of complex systems. Social and biological processes such as the dynamics of epidemics, the diffusion of information in social media, the interactions between species in ecosystems or the communication between neurons in our brains are all actively studied using dynamical models on complex networks. In all of these systems, the patterns of connections at the individual level play a fundamental role on the global dynamics and finding the most important nodes allows one to better understand and predict their behaviors. An important research effort in network science has therefore been dedicated to the development of methods allowing to find the most important nodes in networks. In this short entry, we describe network centrality measures based on the notions of network traversal they rely on. This entry aims at being an introduction to this extremely vast topic, with many contributions from several fields, and is by no means an exhaustive review of all the literature about network centralities.
The propagations of diseases, behaviors and information in real systems are rarely independent of each other, but they are coevolving with strong interactions. To uncover the dynamical mechanisms, the evolving spatiotemporal patterns and critical phenomena of networked coevolution spreading are extremely important, which provide theoretical foundations for us to control epidemic spreading, predict collective behaviors in social systems, and so on. The coevolution spreading dynamics in complex networks has thus attracted much attention in many disciplines. In this review, we introduce recent progress in the study of coevolution spreading dynamics, emphasizing the contributions from the perspectives of statistical mechanics and network science. The theoretical methods, critical phenomena, phase transitions, interacting mechanisms, and effects of network topology for four representative types of coevolution spreading mechanisms, including the coevolution of biological contagions, social contagions, epidemic-awareness, and epidemic-resources, are presented in detail, and the challenges in this field as well as open issues for future studies are also discussed.
Searching for influential spreaders in complex networks is an issue of great significance for applications across various domains, ranging from the epidemic control, innovation diffusion, viral marketing, social movement to idea propagation. In this paper, we first display some of the most important theoretical models that describe spreading processes, and then discuss the problem of locating both the individual and multiple influential spreaders respectively. Recent approaches in these two topics are presented. For the identification of privileged single spreaders, we summarize several widely used centralities, such as degree, betweenness centrality, PageRank, k-shell, etc. We investigate the empirical diffusion data in a large scale online social community -- LiveJournal. With this extensive dataset, we find that various measures can convey very distinct information of nodes. Of all the users in LiveJournal social network, only a small fraction of them involve in spreading. For the spreading processes in LiveJournal, while degree can locate nodes participating in information diffusion with higher probability, k-shell is more effective in finding nodes with large influence. Our results should provide useful information for designing efficient spreading strategies in reality.
Algorithms for community detection are usually stochastic, leading to different partitions for different choices of random seeds. Consensus clustering has proven to be an effective technique to derive more stable and accurate partitions than the ones obtained by the direct application of the algorithm. However, the procedure requires the calculation of the consensus matrix, which can be quite dense if (some of) the clusters of the input partitions are large. Consequently, the complexity can get dangerously close to quadratic, which makes the technique inapplicable on large graphs. Here we present a fast variant of consensus clustering, which calculates the consensus matrix only on the links of the original graph and on a comparable number of additional node pairs, suitably chosen. This brings the complexity down to linear, while the performance remains comparable as the full technique. Therefore, our fast consensus clustering procedure can be applied on networks with millions of nodes and links.
Identifying the node spreading influence in networks is an important task to optimally use the network structure and ensure the more efficient spreading in information. In this paper, by taking into account the shortest distance between a target node and the node set with the highest $k$-core value, we present an improved method to generate the ranking list to evaluate the node spreading influence. Comparing with the epidemic process results for four real networks and the Barab{a}si-Albert network, the parameterless method could identify the node spreading influence more accurately than the ones generated by the degree $k$, closeness centrality, $k$-shell and mixed degree decomposition methods. This work would be helpful for deeply understanding the node importance of a network.
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